Higher petrol prices are seen driving a pickup in inflation to 3.0% in June from 2.7% in May, according to the average in a Reuters poll of economists. But some see inflation going as high as 3.2% – meaning it is more than one percentage point above the Bank's government-set target of 2% and that the governor must write to George Osborne alongside minutes on 14 August to the Bank's next policy meeting.

Penning such a letter would be an especially frustrating task for Carney given he took the helm from Mervyn King on 1 July so can hardly be blamed for target-busting inflation in June. Still, it would fall to him to explain why living costs are so high.

Whether inflation hits 3.1% is "a close run thing", says Alan Clarke, economist at ScotiaBank in London. He points out that petrol prices fell by around 3% on the month in June 2012 and explains that setting that against a 0.8% rise this June adds almost 0.2 percentage points to headline annual inflation.

Victoria Clarke at Investec also sees upward pressure on inflation from petrol as well as food and clothing prices. None should be "quite significant enough to push inflation up to that letter writing mark", she says. But she adds: "We should caution that the airfares wild card remains a threat as ever, with the moving timing of Easter having swayed recent monthly releases making it trickier to benchmark forecast moves off recent months."

Annual inflation has been running persistently above the Bank of England's 2% target for more than three years and King was forced to write a total of 14 explanatory letters to chancellors, starting with Gordon Brown in 2007.

Bank policymakers have made it clear that they will take a flexible approach to inflation targeting if the broader economic picture requires it, potentially paving the way for more quantitative easing – pumping cash into the economy by acquiring sovereign debt from financial institutions – despite above-target inflation.

Bank watchers will get a clearer steer on where former Bank of Canada chief Carney stands when the minutes from his first monetary policy committee meeting are published on Wednesday. They will sum up the discussion and most interestingly, perhaps, show how the nine committee members voted.

On the whole, economists expect the vote against more QE in July was seven members to two, compared with 6-3 at King's final meeting in June when he joined Paul Fisher and David Miles in calling for £25bn more in asset purchases.

Howard Archer, economist at IHS Global Insight said: "Major attention will be focused on the minutes of the July meeting of the Bank of England's monetary policy committee, with the million (Canadian) dollar question being did new Bank of England governor Mark Carney favour more quantitative easing?

"And if he was in favour of more QE, are there signs that he is getting other MPC members to come round to that view as well? It is actually hard to see how the six MPC members that have repeatedly voted against more QE since a split vote first appeared in February would have changed their mind at the July meeting given the recent overall markedly improved news on the economy."

Carney's first policy meeting ended as expected with interest rates sticking at their record low of 0.5% and no further QE on top of the £375bn so far. But the new governor and his colleagues surprised markets with a statement alongside the decision that there was no need for the recent sharp rise in yields on government bonds. It was only the sixth time in its 16-year history that the MPC had issued a statement alongside a decision to leave policy unchanged.